
With the use of tables [1]}, it is
readily shown that for \(n\)  and \(q\)  both odd or both even, the integral
\(\int _{0}^{2\pi }\mathrm {e}^{Jn\psi }J_{q}(\alpha \cos \psi )\,&=2\pi J_{(q-n)/2}(\alpha /2)J_{(q+n)/2}(\alpha /2),\) 
